WIEFERICHS AND THE PROBLEM *(p2) = z(p)

JOSEPH J. HEED Norwich University, Northfield, VT 05663

(Submitted January 1982)

INTRODUCTION

1. Let z{ri) be the index of the first Fibonacci number divisible by the natu-ral number n. At this writing, there has not been found a prime p whose square enters the Fibonacci sequence at the same index as does p. This does not occur for p < 106 [2],

The problem is related to the following one. For what relatively prime p, b9 is it true that p2\bp~1 - 1? Apparently, this question was first asked by Abel. Dickson [1] devotes a chapter to related results. For b = 2, the con-forming p 2 values are the well-known Wieferich squares, which enter in the solu-tion of Fermat's Last Theorem. The only two Wieferich squares with p < 3 109

are 10932 and 35112 [6, p. 229]. These phenomena are rare but, to a degree, predictable. An investigation of this predictability sheds some light on the Fibonacci phenomenon.

2.1 Notation. Define n\bx - 1 as meaning n\bx - 1, and n\by - 1 for y < x (i.e., b belongs to the exponent x> modulo ri) .

2.2 The following are well known. For p prime, (b, p) = 1; if p\\ba - 1, then p\bB - 1 if and only if 3 = k a. Since p\bp x - 1 (Fermat), it follows that a\p - 1. For q prime, (b, q) = 1; if q\\by - 1, then pq\\blcm(a'Y) - 1 . The mul-tiplicative properties are similar to those of the Euler 3, such that 1/p2 has a period the same length as 1/p, i.e., p2|10p~ 1, is 487. Its period is of length 486.) It can be shown that this deviation oc-curs if and only if p 2 | & p _ 1 - 1. If such is the case, and imitating Shanksfs flair for coinage of such terms, we say p is a wieferich, modulo b.

2.3 Consider the solutions to xp~1 E 1 (mod p 2 ) . Gauss [3, art. 85] assures us that there are p - 1 distinct solutions, xs between 1 and p 2 - 1.

For each b, 1 < b < p, there is a distinct k such that

(b + kp^'1 = 1 (mod p 2 ) .

These provide the p - 1 solutions:

(b + kp)p~1 - 1 = bp~x - 1 + (p - l)bp~2kp (mod p2) and

(- M - b~Xk = 0 (mod p ) , yielding k = b{~ ) (mod p) .

If x is a solution, so too is p - x. x = 1 is always a solution; there-fore, (p - 3)/2 solutions are scattered from x = 2 to x = (p2 - l)/2. If ran-domly distributed, the probability that a particular x = b is a solution is

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VIEFERICHS AND THE PROBLEM s(p2) = z(p)

(p - 3)/(p2 - 3). Holding b fixed and letting p range, the expected number of solutions encountered < P is Ep(p - 3)/(p2 - 3). Since the series is divergent (p

WIEFERICHS AND THE PROBLEM s(p2) = z (p)

From 3.1, q ^ c2 (mod p), so p|^fea and, thus, p2\Tr. But this contradicts the hypothesis that a was the smallest such index.

3.2.2 Lemma: If p||Ta, thenp2|rpa. Consider:

Noting that Rp~1 is an integer, P-i

i?^ 1 ^)^ =^ ^JL-+- (-1)*(OT)*"(P)[^ ^ -J.

yPa _ lJFPa p d i v i d e s a l l terms bu t = Tpa , so i t must d i v i d e i t a l s o .

3 . 2 . 3 Lemma: I f p\\Ta bu t p 2 | | r t a , 1 < t < p , t h e n , s i n c e p2\Vkta (from 3 . 2 . 1 ) and p2\Tpa (from 3 . 2 . 2 ) , i t fo l lows t h a t t\p; bu t p i s p r ime , so

p2| |ra or p 2 | | r p a . In the former case, p\Tp1; in the latter, since pi 1 is not a multiple of pa, p2|rp 1. This establishes the result.

3.3 We next consider , with c = c + E,p and q = q1 + p, expand and reduce \j/Pi _ lpi

5 (mod p ). The result is linear in and . Thus, for given c, q, yPi _ ypi

for = = 0 (mod pA) , each , 0 < ^ < p, generates one , 0 < < p.

Fix e. Let q range from 1 to (p - 1). One of these pairs (c, q), that with q= c2 (mod p) , will produce a sequence not containing an entry point for p [4]. The other p- 2 pairs will each generate a solution = 0, = 0 yielding a se-quence with associated with c + / 106. On the other hand, In In 106 is not yet 3, perhaps not too wide a miss?

REFERENCES

1. L. E. Dickson. History of the Theory of Numbers, Vol. I. New York: Chelsea Publishing Company, 1952.

2. L. A. G. Dresel. Letter to the Editor. The Fibonacci Quarterly 15. no. 4 (1977):346.

3. C. F. Gauss. Disquisitiones Arithneticae. New Haven: Yale Univ. Press, 1966.

4. J. J. Heed & L. Kelly. "Entry Points of the Fibonacci Sequence and the Euler cj) Function.11 The Fibonacci Quarterly 16, no. 1 (1978) :47.

5. H. Rademacher. Lectures on Elementary Number Theory. New York: Blaisdell Publishing Company, 1964.

6. D. Shanks. Solved and Unsolved Problems in Number Theory, 2nd ed. New York: Chelsea Publishing Company, 1978.

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